Yes, Riley Cooper should face some sort of punishment for using a racial epithet. Cooper is in the American spotlight on a weekly basis, and as such, he serves as a role model for many children. His use of a terrible word is not a personal matter, but a cultural one.
I would punish him, no questions asked. Racism cannot be tolerated in this day and age. But the answer to the question is based on what the NFL and the Philadelphia Eagles feel is necessary. These organizations do not want someone who makes racial slurs representing them. It damages their brand and player relationships. So as for my personal view, he should absolutely be punished. In terms of punishment at the organizational level, he should be punished because he represents his employer.